FractionalGeometry-Chap2

While processor speed and memory are not such big

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Unformatted text preview: [−2, −1] and B = [1, 2]. Then for all δ > 0, A ∩ B = ∅ and so (A ∩ B )δ = ∅. On the other hand, for δ > 1, Aδ ∩ Bδ = (1 − δ, δ − 1). Exercises Prob 2.3.1 Take A = {(0, y ) : 0 ≤ y ≤ 1} and B = {(x, 0) : 0 ≤ x ≤ 1}, and T the IFS T1 (x, y ) = (x/2, y/2), and for i = 2, 3 Ti (x, y ) = (x/4, y/4) + (ei , fi ) 50 CHAPTER 2. ITERATED FUNCTION SYSTEMS with (e2 , f2 ) = (3/4, 0) and (e3 , f3 ) = (0, 3/4). For these sets and IFS, (a) Compute h(A, B ). (b) Compute the Euclidean contraction factors of T1 , T2 , and T3 . (c) Compute h(T (A), T (B )). (d) Reconcile this result with Prop. 2.3.2. Prob 2.3.2 From the proof of Lemma 2.3.1 (a) Show Wδ ∪ Xδ ⊆ (W ∪ X )δ . (b) Does Wδ ∪ Xδ = (W ∪ X )δ for any W and X ? (c) Does Wδ ∪ Xδ = (W ∪ X )δ for all W and X ? Prob 2.3.3 Denote by G the right Sierpinski gasket with vertices (0, 0), (1, 0), and (0, 1). Recall the gasket transformations are T1 (x, y ) = (x/2, y/2), T2 (x, y ) = (x/2, y/2) + (1/2, 0)...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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